Interlaced dense point and absolutely continuous spectra for Hamiltonians with concentric-shell singular interactions

TL;DR

This paper investigates the spectral properties of a Schrödinger operator with concentric-shell singular interactions, revealing interlaced dense point and absolutely continuous spectra, with implications for high-energy behavior and eigenvalue distribution.

Contribution

It demonstrates the interlacing of dense point and absolutely continuous spectra for such operators and analyzes how interaction parameters influence spectral segments at high energies.

Findings

Essential spectrum consists of interlaced dense point and absolutely continuous segments.

High-energy spectral segment lengths depend on interaction parameters.

For two-dimensional case, an infinite number of eigenvalues exist below the lowest spectral band.

Abstract

We analyze the spectrum of the generalized Schrodinger operator in $L^2(R^\nu) \nu \geq 2$, with a general local, rotationally invariant singular interaction supported by an infinite family of concentric, equidistantly spaced spheres. It is shown that the essential spectrum consists of interlaced segments of the dense point and absolutely continuous character, and that the relation of their lengths at high energies depends on the choice of the interaction parameters; generically the p.p. component is asymptotically dominant. We also show that for $\nu=2$ there is an infinite family of eigenvalues below the lowest band.